# A real number as a point

1. Jul 15, 2015

### shuxue

An element of the domain of a real-valued function of a real variable is often called a point. For example, an element (point) $p$ in the domain of a real-valued function $f$ of a real variable where $f'(p)=0$ or $f'(p)$ is undefined is called a critical point of the function. The particular type of critical point $x$ where $f'(x)=0$ is called a stationary point. As another example, "the graph $y=(x+2)(x-1)^2$ cross the x-axis at the points $x=-2$ and $x=1$." In those cases, why we called a real number as a point? Is it because we view the real numbers as points in the context of the real line?

2. Jul 15, 2015

### tommyxu3

On the graph I think I should say it cross $x$-axis at the points $(-2,0)$ and $(1,0).$ I think this is more specfic, but I'm not sure whether your statement is wrong or not.
I think your former example's reason is that the whole $x-y$ plane is the domain of the function. They are also true points, aren't they?

3. Jul 16, 2015

### Staff: Mentor

Each point on a number line represents a real number.

In shuxue's example, it was clearly stated that the points were on the x-axis, so the point on the x-axis where x = -2 is also the ordered pair (-2, 0).
No, in the first example, it says that the function is of one variable, so the domain is some subset of the real numbers, possibly including the entire real number line.
???
I don't understand what this refers to.